Planck-Scale Corrections to Friedmann Equation

Cent. Eur. J. Phys. • 12(4) • 2014 • 245-255 DOI: 10.2478/s11534-014-0441-3

Central European Journal of Physics

Planck-scale corrections to Friedmann equation

Research Article

Adel Awad∗1,2,3, Ahmed Farag Ali† 1,4

1 Centre for Fundamental Physics, Zewail City of Science and Technology Sheikh Zayed, 12588, Giza, Egypt 2 Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt 3 Center for Theoretical Physics, British University of Egypt, Sherouk City 11837, P.O. Box 43, Egypt 4 Department of Physics, Faculty of Science, Benha University, Benha 13518, Egypt

Received 16 November 2013; accepted 18 January 2014

Abstract: Recently, Verlinde proposed that gravity is an emergent phenomenon which originates from an entropic force. In this work, we extend Verlinde’s proposal to accommodate generalized uncertainty principles (GUP), which are suggested by some approaches to quantum gravity such as string theory, black hole physics and doubly special relativity (DSR). Using Verlinde’s proposal and two known models of GUPs, we obtain modiﬁcations to Newton’s law of gravitation as well as the Friedmann equation. Our modiﬁcation to the Friedmann equation includes higher powers of the Hubble parameter which is used to obtain a corresponding Raychaudhuri equation. Solving this equation, we obtain a leading Planck-scale correction to Friedmann-Robertson-Walker (FRW) solutions for the p = ωρ equation of state. PACS (2008): 04.60.Bc;04.60.Cf;04.70.Dy Keywords: quantum gravity • entropic force • thermodynamics • black holes © Versita sp. z o.o.

1. Introduction mutation relations between position coordinates and momenta [1–8]. This can be understood in the context of string theory since strings can not interact at distances smaller than their size. The GUP is represented by the following One of the intriguing predictions of various frame works form [9–11]: of quantum gravity, such as string theory and black ~ hole physics, is the existence of a minimum measurable ∆xi∆pi ≥ [1 + β (∆p)2+ < p >2 2 length. This has given rise to the so-called generalized un- 2 2 + 2β ∆pi + < pi > ] ; (1) certainty principle (GUP) or equivalently, modiﬁed com- `2 p2 P p p β β / M c 2 β p M where = j j , = 0 ( p ) = 0 2 , p = Planck ∗ j ~ E-mail: [email protected] † E-mail: [email protected] (Corresponding author) mass, and Mpc2 = Planck energy. The inequality (1) is

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equivalent to the following modified Heisenberg algebra scales are only a few orders of magnitude less than Planck [9]: scale. [xi; pj ] = i~(δij + βδij p2 + 2βpipj ) : (2) Let us review Verlinde’s proposal [38] on the origin of gravity, where he suggested that a gravitational force might This form ensures, via the Jacobi identity, that be of an entropic nature. This assumption is based on the x ; x p ; p [ i j ] = 0 = [ i j ] [10]. relation between gravity and thermodynamics [40–42]. Ac- cording to thermodynamics and the holographic principle, Recently, a new form of GUP was proposed in [12, Verlinde’s approach results in Newton’s law of gravitation, 13], which predicts maximum observable momenta as well the Einstein Equations [38] and the Friedmann Equations as the existence of minimal measurable length, and is con- [43]. The theory can be summarized as sistent with doubly special relativity theories (DSR) [14– at the temperature T , the entropic force F of a gravita- 19], string theory and black holes physics [1–7]. It ensures tional system is given as: [xi; xj ] = 0 = [pi; pj ], via the Jacobi identity. F∆x = T ∆S; (4) h p p x ; p i δ − α pδ i j [ i j ]= ~ ij ij + p where ∆S is the change in the entropy such that, at a i x + α2 p2δij + 3pipj ; (3) displacement ∆ , each particle carries its own portion of entropy change. From the correspondence between the entropy change ∆S and the information about the bound- α α /Mpc α `p/~;Mp `p where = 0 = 0 = Planck mass, = ary of the system and using Bekenstein’s argument [40– Mpc2 Planck length, and = Planck energy. In a series 42], it is assumed that ∆S = 2πkB, where ∆x = ~/m and of papers, various applications of the new model of GUP kB is the Boltzmann constant. have been investigated [20–33]. mc The upper bounds on the GUP parameter α has been ∆S = 2πkB ∆x; (5) derived in [22]. It was suggested that these bounds can ~ be measured using quantum optics and gravitational wave where m is the mass of the elementary component, c is techniques in [34, 35]. Recently, Bekenstein [36, 37] pro- speed of light and ~ is the Planck constant. posed that quantum gravitational effects could be tested The holographic principle assumes that for a region en- experimentally, suggesting“a tabletop experiment which, closed by some surface gravity can be represented by the given state of the art ultrahigh vacuum and cryogenic degrees of freedom on the surface itself and independent technology, could already be sensitive enough to detect of the its bulk geometry. This implies that the quantum Planck scale signals” [36]. This would enable several gravity can be described by a topological quantum field quantum gravity predictions to be tested in the Laboratory theory, for which all physical degrees of freedom can be [34, 35]. This is considered as a milestone in the field of projected onto the boundary [44]. The information about quantum gravity phenomenology. the holographic system is given by N bits forming an ideal Motivated by the above arguments, we investigate pos- gas. It is conjectured that N is proportional to the entropy sible effects of GUP on the Friedmann equation. We use of the holographic screen, the entropic force approach suggested by Verlinde [38] to calculate corrections to Newton’s law of gravitation and the Friedmann equations for two models of GUP, which 4S N = ; (6) are mentioned above. We found that Planck-scale cor- kB rections to the Friedmann equation include higher powers of the Hubble parameter which are suppressed by then according to Bekenstein’s entropy-area relation [40– Planck length. Using these corrections we construct the 42] corresponding Raychaudhuri equations, which we then kBc3 S = A: (7) solve to obtain a leading Planck-scale correction to the 4G~ Friedmann-Robertson-Walker (FRW) solutions with equa- Therefore, one gets tion of state p = ωρ. Deriving the effects of GUPs on Newton’s law of gravitation through the entropic force Ac3 4πr2c3 N = = ; (8) approach was initially reported in [39] where a modi- G~ G~ fied Newton’s law of gravitation was calculated. A possible application of these Planck-scale corrections is early where r is the radius of the gravitational system and A = cosmology, in particular inflation models where physical 4πr2 is area of the holographic screen. It is assumed that

246 Brought to you by | CERN library Authenticated Download Date | 10/4/17 1:58 PM Adel Awad, Ahmed Farag Ali each bit emerges out of the holographic screen i.e. in one 2. GUP-quadratic in ∆p and BH dimension. Therefore each bit carries an energy equal thermodynamics to kBT/2. Using the equipartition rule to calculate the energy of the system, one gets In this section, we review the modified thermodynamics of the black hole which yields a modified entropy due to πc3r2 E 1 Nk T 2 k T Mc2: GUP [56–62]. Using the holographic principle, we get a = B = G B = (9) 2 ~ modified number of bits N which yields quantum gravity corrections to Newton’s law of gravitation and the Fried- By substituting Eq. (4) and Eq. (5) into Eq. (9), we get mann equations. Black holes are considered as a good laboratories for the Mm clear connection between thermodynamics and gravity, so F G ; = r2 (10) black hole thermodynamics will be analyzed in this section. We then make an analysis of BH thermodynamics p making it clear that Newton’s law of gravitation can be if the GUP-quadratic in ∆ that was proposed in [1–11] derived . is taken into consideration. With Hawking radiation, the emitted particles are mostly photons and standard model Recently, a modified Newton’s law of gravity due to (SM) particles. Using the Hawking-Uncertainty Relation, Planck-scale effects through the entropic force approach the characteristic energy of the emitted particle can be was derived by one of the authors [39]. Derivation of identified [56–60]. It has been found [25, 26], assuming Planck scale effects on the Newton’s law of gravity are some symmetry conditions from the propagation of Hawk- based on the following procedure: modified theory of grav- ing radiation, that the inequality that would correspond ity → modified black hole entropy→ modified holographic to Eq. (1) can be written as follows: surface entropy → Newton’s law corrections . This proce- ~ p2 x p ≥ 5 µ β `2 ∆ : dure has been followed with other approaches like non- ∆ ∆ 1 + (1 + ) 0 p (11) 2 3 ~2 commutative geometry in [45–48]. In this paper, we take 2:821 2 into account the quantum gravity corrections due to GUP where µ = π . in the entropic-force approach, following the same proce- By solving the inequality (11) as a quadratic equation in dure as [39], and extend our study to calculate the modified ∆p, we obtain Friedmann and Raychaudhuri equations. To calculate the quantum gravity corrections to the Friedmann equations,  s  we use the procedure that has been followed in [49, 50]. It 5 µ β `2 p x (1 + ) 0 p ∆ ≥ ∆  − − 3  : is worth mentioning that recently, there have been some 5 µ β `2 1 1 x2 ~ (1 + ) 0 p ∆ considerable interest in entropic force approach and its 3 applications [51–55]. (12) An outline of this paper is as follows. In Sec .2, We Using Taylor expansion, Eq. (12) reads " # investigate entropic force if the GUP of Eq. (1), which µ β `2 (1 + ) 0 p was proposed earlier by [1–11], is taken into considera- p ≥ 1 5 O β2 ; ∆ x 1 + x2 + ( 0 ) (13) tion. First we calculate the modified thermodynamics of 2∆ 12 ∆ the black hole which yields a modified entropy. By the where we used natural units as ~ = 1. According to [61, holographic principle, we calculate the modified number of N 62] a photon is used to ascertain the position of a quantum bits . Based on the modified number of bits, we estimate particle of energy E and according to the argument in the Planck scale correction to Newton’s law of gravitation [63, 64] which states that the uncertainty principle ∆p ≥ and the Friedmann equations. We solve Raychaudhuri 1/∆x can be translated to the lower bound E ≥ 1/∆x, one equations to obtain Planck scale corrections to FRW cos- can write for the GUP case mological solutions. " # µ β `2 In Sec. 3, we repeat the analysis for the newly pro- 1 5 (1 + ) 0 p E ≥ 1 + + O(β2) (14) posed model of GUP in Eq. (3). We calculate different 2∆x 12 ∆x2 0 corrections to Newton’s law of gravity and the Friedman equations. We compare our results with previous stud- For a black hole absorbing a quantum particle with energy ies of quantum gravity corrections to gravity laws. We E and size R, the area of the black hole should increase then solve the modified Raychaudhuri equations. Further by [65, 66]. 2 implications are discussed. ∆A ≥ 8π `p ER; (15)

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The quantum particle itself implies the existence of ﬁnite bound given by κ dA T = 8π dS A ≥ π `2 E x: " # ∆ min 8 p ∆ (16) µ β `2 κ 5 π (1 + ) 0 p = 1 + + O(β2) : (23) 8π 48 A 0 Using the Eq. (14) in the inequality (16), we get

" # So far, it is concluded that the temperature is not only 2 µ β `p 2 5 (1 + ) 0 2 proportional to the surface gravity but also it depends on ∆Amin ≥ 4π `p 1 + + O(β ) : (17) 12 ∆x2 0 the black hole’s area [61, 62].

The value of ∆x is taken to be inverse of surface gravity 2.1. Modified Newton’s law of gravity − ∆x = κ 1 = 2rs where rs is the Schwarzschild radius. In this section we study the implications of the correc- Where this is probably the most sensible choice of length tions for the entropy in Eq. (59) and calculate how the scale is in the context of near-horizon geometry [61, 62]. number of bits of Eq. (6) would be modified leading to This implies that A a new correction to Newton’s law of gravitation. Using x2 : ∆ = π (18) the corrected entropy given in Eq. (22), we find that the number of bits should be corrected as follows: Substituting Eq. (18) into Eq. (19), we get

! ! S A π µ A " # 0 4 5 (1 + ) 2 N − β : π µ β `p = = 2 0 ln 2 (24) 2 5 (1 + ) 0 2 kB `p 12 `p ∆Amin ' λ`p 1 + + O(β ) : (19) 12 A 0 √ ` ~ G We can substitute for the Planck length with p = c3/2 , where the parameter λ will be defined later. According to then we get the modified number of bits as follows [40–42], the black hole’s entropy is conjectured to depend S A c3 π µ A c3 on the horizon’s area. From information theory [67], it has N0 4 − 5 (1 + ) β : = = 0 ln (25) been found that the smallest increase in entropy should be kB ~ G 12 ~ G independent of the area. It is just one bit of information, b δ 5 π (1+µ) β which is = ln(2). We define the constant = 12 0, then we have

S A c3 A c3 N0 4 − δ : dS Smin b = k = G ln G (26) ∆ ; B ~ ~ dA = A = h π µ β `p2 i (20) ∆ min λ`2 5 (1+ ) 0 O β2 p 1 + A + ( 0 ) 12 By substituting Eq. (26) into Eq. (9) and using Eq. (4), we get where b is a parameter. By expanding the last expression β πc3 ! in orders of 0 and then integrating, we get the entropy F r2 c2 4 r2 E Mc2 − δ ln ( ~G ) : = = 1 π c3 (27) " !# m G 2 r2 b A π µ b A ~G S − 5 (1 + ) β : = λ `2 0 λ ln `2 (21) p 12 p This implies a modiﬁed Newton’s Law of gravity given as   Using the Hawking-Bekenstein assumption [40–42], which 4πr2 ln ( `2 ) relates entropy with the area, the value of b/λ is ﬁxed to GMm p F = 1 + δ  : (28) be = 1/4, so that r2 4πr2/`p2 ! ! A π µ A S − 5 (1 + ) β : Which means that the Newtonian gravitational potential = `2 0 ln `2 (22) 4 p 48 p would be:   It is concluded that the entropy is directly related to 4πr2 GMm ln ( `p2 ) the area and gets a logarithmic correction when apply- V r −  2 δ 1 1 δ  : ( ) = r 1 + πr2/`2 + πr2/`2 ing GUP approach[61, 62]. 9 4 p 3 4 p The temperature of the black hole is (29)

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We note that these logarithmic corrections to the Newto- Equation (34) can be written in terms of the matter density nian gravitational potential have been obtained using an ρ = M/V;V = (4/3)π˜r3 as: independent approach studying the brane world correc-   tions to Newton’s law of gravity [68] which suggest that 4π˜r2 a πG ln ( `p2 ) brane world and GUP would predict similar physics. In ¨ − 4 ρ  δ  ; a = 1 + πr2/`2 (35) the next subsection, we discuss modification of the Fried- 3 4 ˜ p mann equation due to the corrections of quantum gravity. The last equation is considered to be the modified Newto- 2.2. Modified Friedmann and Raychaudhuri nian cosmology. To derive the modified Friedmann equa- Equations tions, it was assumed as in [43], that the active gravitational mass M, where In this section, we use the analysis of [43, 49, 50, 69] Z which derived the Friedmann equations using an entropic 1 µ ν M = 2 dV Tµν − T gµν u u ; (36) force approach. We investigate the effect of the quantum V 2 gravity arising from the GUP proposed in Eq. (1) on the form of the Friedmann equations using the method of [43]. generates the acceleration rather than the total mass M The Friedmann-Robertson-Walker (FRW) universe is de- in the volume V . scribed by the following metric: So for the FRW universe, the active gravitational mass would be a b π ds2 habdx dx r2d 2; 4 = + ˜ Ω (30) M = (ρ + 3p) a3r3: (37) 3 a M where ˜r = a(t)r; x = (t; r); hab = (−1; a2/(1 − By replacing M with , Eq. (35) can be rewritten as: kr2)); dΩ2 = dθ2 + sin2 θdφ2 and a; b = 0; 1. k is  −1 ; ; − 4π˜r2 the spatial curvature and it takes the values 0 1 1 2 aa¨ ln ( `p2 ) for a flat, closed and open universe, respectively. The M = − r3 1 + δ  : (38) G πr2/`p2 dynamic apparent horizon is determined by the relation 4 ˜ ab h ∂a˜r∂b˜r = 0, which yields its radius[69]: By Equating Eq. (37) with Eq. (38), we get 1 ˜r = a r = √ ; (31) 2 2   H + k/a 4π˜r2 a¨ 4πG ln ( `p2 ) = − (ρ + 3p) 1 + δ  : (39) a 3 4π˜r2/`p2 where H = a/a˙ is the Hubble parameter. By assuming that the matter which occupies the FRW universe forms a perfect fluid, the energy-momentum tensor would be: Using the continuity equation of Eq. (33) in Eq. (39), multiplying both sides with (a a˙and integrating, we obtain Tµν = (ρ + p)uµuν + pgµν : (32) the following equations:

  4π˜r2 The energy conservation law then leads to the continuity 4πG ρ˙ ln ( `p2 ) a˙a¨ = − aa˙ ρ − − 3ρ 1 + δ  : equation 3 H 4π˜r2/`p2 ρ H ρ p ; ˙ + 3 ( + ) = 0 (33) (40)   4π˜r2 To calculate the quantum gravity corrections to the Fried- d 8πG d ln ( `p2 ) (a˙2) = ρa2 1 + δ  : (41) mann equations, we consider a compact spatial region with dt 3 dt 4π˜r2/`p2 volume V and radius ˜r = a(t)r [43]. By combining New- ton’s second law for the test particle m near the surface Integrating both sides for each term of the last equation with the modiﬁed gravitational force of Eq. (28), we get yields

    4π˜r2 Z 4πa2r2 GMm ln ( `p2 ) 8πG δ ln ( `p2 ) F = m ˜¨r = m a¨(t) r = − 1 + δ  : a˙2 + k = ρa2 1 + d ρa2  : ˜r2 4π˜r2/`p2 3 ρa2 4πa2r2/`p2 (34) (42)

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We can take r outside the integration because it does not is a constant of time, so the continuity equation (33) is depend on a from the deﬁnition of ˜r = ra(t) after Eq. (30), integrated to give we get

" πG 2 8 2 a˙ + k = ρa 1 ρ ρ a−3(1+w); 3 = 0 (44) ! # 2 Z δ `p 4πa2r2 d(ρa2) + ln : (43) 4πρa2r2 `p2 a2 By substituting Eq. (44) into Eq. (43) and solving the in- Suppose that the equation of state parameter w = p/ρ tegration, we get

√ ! δ ρ `2 − − ω Z k 8πG 2 0 p ( 1 3 ) 2 πr − − ω H2 + = ρ 1 + ln a a 4 3 da : (45) a2 3 4πρa2r2 `p

The integration in the last equation can be solved to yield " √ # 2 k 8πG δ `p (1 + 3ω) 2 π r a H2 + = ρ 1 + 1 + 3(1 + ω) ln : (46) a2 3 18π˜r2(1 + ω)2 `p

Using the relation ˜r = a r = √ 1 , we get H2+k/a2

" !# 2 2 k δ`p (1 + 3ω) k 3(1 + ω) k `p 8πG H2 + 1 − (H2 + ) 1 − ln H2 + = ρ; (47) a2 18π(1 + ω)2 a2 2 a2 4π 3

or it can be written as " !# 2 2 2 k δ `p (1 + 3ω) k δ `p (1 + 3ω) k k `p 8πG H2 + 1 − (H2 + ) + (H2 + ) ln (H2 + ) = ρ: (48) a2 18π(1 + ω)2 a2 12π(1 + ω) a2 a2 4π 3

The last equation gives the entropy-corrected Friedmann H˙ and H. Since we have H = a/a˙ , then equation of the FRW universe by considering gravity as an entropic force. Notice that the Planck-scale correc- a H˙ −H2 ¨ : tion terms include higher powers of H suppressed by the = + a (49) Planck scale, the only physical scale we have. We then derive the Raychaudhuri equation which corre- Substituting Eq. (39) and Eq. (48) into Eq. (49) and using sponds to Eq. (48). We need to ﬁnd the relation between ω = p/ρ, we get:

2 2 3(1 + ω) 1 + 3ω k δ`p (1 + 3ω)2 k H˙ = − H2 − + H2 + 2 2 a2 18π 2(1 + ω) a2 ! 2 2 2 δ`p 1 + 3ω k `p k + H2 + ln H2 + : (50) 4π 3(1 + ω) a2 4π a2

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−1 2.3. Solutions of the modified Raychaudhuri point at H ∼ `p (see for example [70]), i.e. a de Sit- equation ter space, which smoothes the big bang singularity. As one might expect, the approximation used so far, H`p < 1, In this section we solve the modified Raychaudhuri equa- breaks down around the fixed point and we need an exact k tion (50) for the flat case ( = 0), using a perturbation treatment to describe the solution near this point. There- k method. For = 0, equation (50) has the form fore, one can only trust a perturbative solution where H`p < 1 which we write as

2 ω δ`p ω 2 3(1 + ) 2 (1 + 3 ) 4 H t H λH H˙ = − H + H ( ) = 0 + 1 (52) 2 36π (1 + ω) " # ` H 6 p λ δ`2 H t / γ t C × √ : where = p and 0( ) = 1 ( + 1) is the solution of 1 + ω ln π (51) 1 + 3 4 Eq. (49) when λ = 0. Solving Eq. (49) up to a leading order in λ and imposing It is interesting to observe that this equation has a ﬁxed initial conditions we get

" # H β H 3ξ σ/ξ γH t − t − H t 0 λ 0 ( ) ln ( 0( 0) + 1) 1 ; ( ) = γH t − t + γ γH t − t 2 1 + γH t − t (53) ( 0( 0) + 1) ( 0( 0) + 1) ( 0( 0) + 1)

H t γ where 0 is the initial Hubble parameter at time 0, = The scale factor can be calculated using the relation / ω β ω 2/ π ω σ / ω H t a/a λ 3 2(1 + ), = (1 +√ 3 ) [36 (1 + )], = 6 (1 + 3 ) ( ) = ˙ , and it is given to the ﬁrst order of as ξ σ ` H / π − σ and = ln p 0 4 + 1 . follows:

" !# β H2η σ/η γH t − t − a t a γH t − t 1/γ − λ 0 ln ( 0( 0) + 1) 1 ; ( ) = 0 ( 0( 0) + 1) 1 γ2 γH t − t 1 + γH t − t (54) 2 ( 0( 0) + 1) ( 0( 0) + 1)

√ η σ ` H / π − / σ p where = ln p 0 4 + 1 3 2 . ∆ . It has been found in [25, 26], that the inequality which Although, it is not clear from the approximation we have would correspond to Eq. (3) can be written as follows: here if these corrections are going to resolve big bang singularities or not, a possible application of these corrections is early cosmology, in particular inﬂation models where physical scales are only a few orders of magnitude " √ p less than Planck scale. x p ≥ ~ − α ` 4 µ ∆ ∆ ∆ 1 0 p 2 3 ~ # p2 2 2 ∆ 3. GUP linear and quadratic in p + 2 (1 + µ) α `p : (55) ∆ 0 ~2 and BH thermodynamics

In this section, we will repeat the same analysis that was done in section 2, but with GUP linear and quadratic in Solving it as a quadratic equation in ∆p results in

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√  v  p x α ` 4 µ u µ α2`2 ∆ 2∆ + 0 p u 8 (1 + ) 0 p ≥ 3  − t − √  : µ α2 `2 1 1 2 (56) ~ 4 (1 + ) p x α `p 4 µ 0 2∆ + 0 3

The negatively-signed solution is considered as the It is apparent, that Eq. (62) implies a modification to New- one that refers to the standard uncertainty relation as ton’s law of gravitation [39]; `p/ x → ∆ 0. Using the Taylor expansion, we find that √ Mm α µ F = G 1 − : (63) √ r2 3 r p ≥ 1 − 2 α ` µ 1 : ∆ 1 0 p (57) ∆x 3 ∆x This equation states that the modification in Newton’s We repeat the same analysis of Sec. (2), and we get law of gravity seems to agree with the predictions of the Randall-Sundrum II model [71] which contains one un- r compactified extra dimension and length scale ΛR . The 2 µ π Amin λ`2 − α `p ; only difference is the sign. The modification in Newton’s ∆ = p 1 0 A (58) 3 gravitational potential on the brane [72] is given as

Which leads to the modiﬁed entropy as  mM 4ΛR −G r 1 + πr ; r ΛR  3 s VRS = ; (64) A A  S 2 α π µ : −G mM 2ΛR ; r = + 0 (59) r 1 + r2 ΛR 4 `p2 3 4 `p2 3

where r and ΛR are the radius and the characteristic We find that the entropy is directly related to the area length scale respectively. The result, Eq. (63), agrees and is modified when applying GUP-approach. The tem- with Eq. (64), albeit with the opposite sign when r ΛR . perature of the black hole is This result says that α ∼ ΛR which helps to set a new upper bound on the value of the parameter α. This means κ dA κ r π T − 2 α ` µ : that the proposed GUP-approach [12, 13] is apparently = = 1 0 p (60) 8π dS 8π 3 A able to predict the same physics as Randall-Sundrum II model. In recent gravitational experiments, it is found that /r2 We find that the temperature is not only proportional to the Newtonian gravitational force, the 1 -law, seems to ∼ : − : the surface gravity but also it depends on the black hole’s be maintained up to 0 13 0 16 mm [73, 74]. How- area. ever, it is unknown whether this law is violated or not at sub-µm range. Further implications of these modifications 3.1. Modified Newton’s law of gravity have been discussed in [75] which could be the same for the GUP modification. This similarity between the GUP In this section we study the implications of the corrections and the extra dimensions of the Randall-Sundrum II model α calculated for the entropy in Eq. (59) and calculate how would lead to new bounds on the GUP parameter with the number of bits of Eq. (6) would be modified to identify respect to the extra dimension length scale ΛR . Further new corrections to Newton’s law of gravitation. Using investigations are needed. the corrected entropy given in Eq. (59), we find that the number of bits should also be corrected as follows. 3.2. Modified Friedmann and Raychaudhuri s equations S A A N00 4 4 α µ π : = = + 0 (61) kB `p2 3 `p2 In this subsection, we calculate the quantum gravity corrections predicted by the GUP of Eq. (3). we repeat the By substituting Eq. (61) into Eq. (9) and using Eq. (4), we same analysis that was done in subsection (2.2), but this get time using the modified Newton’s law of gravity of Eq. (63). √ r2 α µ r After making the same analysis of subsection (2.2), we E = F c2 + : (62) m G 3 m G get the modified acceleration equation for the dynamic

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evolution of the FRW universe as follows: Similar to the quadratic GUP case, the above equation has −1 √ a fixed point at H ∼ `p . The above equation was studied a πG α µ ¨ − 4 ρ p − : by Murphy [76], where an exact solution was presented. a = ( + 3 ) 1 r (65) 3 3 ˜ Recently, the above equation was studied in [70] as an example to show that pressure properties, such as asymp- Using the continuity equation of Eq. (33) in Eq. (39) and totic behavior and fixed points can be used to qualitatively a a multiplying both sides with ( ˙), we get after making describe the entire behavior of a FLRW solution. Similar integration the following equation: to the quadratic case, the approximation i.e., H`p < 1, √ breaks down around the fixed point and one should have πG α µ Z d ρ a2 2 8 2 1 ( ) an exact treatment to describe the solution near this point. a˙ + k = ρa 1 − : (66) 3 3 ρ a ˜r a Here we are interested in a perturbative solution, which can written as H t H H By substituting Eq. (44) into Eq. (43) and integrating, we ( ) = 0 + 1 (73) get √ α µ H t / γ t C where = 6 and 0( ) = 1 ( + 1) is the solution of √ k 8 π G α µ 1 + 3ω 1 Eq. (49) as we set = 0. H2 + = ρ 1 − : (67) a2 3 3 2 + 3ω ˜r Solving Eq. (49) up to a leading order in and imposing Using the relation ˜r = a r = √ 1 , we get: initial conditions we get H2+k/a2 H " √ r # H t 0 k π G α µ ω k ( ) = γH t − t 2 8 1 + 3 ( 0( 0) + 1) H + = ρ 1 − H2 + ; a2 3 3 2 + 3ω a2 β0 H 2 γH t − t 0 ln ( 0( 0) + 1) (68) + γ γH t − t 2 (74) ( 0( 0) + 1) which can be written as: H t γ " √ r # where 0 is the Hubble parameter at initial time 0, = k α µ ω k 0 H2 1 + 3 H2 3/2(1 + ω), β = (1 + 3ω)/(2 + 3ω). + a2 1 + ω + a2 3 2 + 3 The scale factor can be calculated using the relation 8 π G H(t) = a/a˙ , and it is given to the first order of as = ρ: (69) 3 follows:

We then derive the Raychaudhuri equation which corre- a t a γH t − t 1/γ ( ) = 0 ( 0( 0) + 1) sponds to Eq. (69) in which we ﬁnd the relation between " # β0 H γH t − t H˙ and H. − 0 ln ( 0( 0) + 1) a 1 γ2 γH t − t (75) H˙ −H2 ¨ : ( 0( 0) + 1) = + a (70) Substituting Eq. (65) and Eq. (69) into Eq. (70), and using ω = p/ρ, we get: 4. Conclusions

3(1 + ω) 1 + 3ω k Verlinde has extended the validity of the holographic prin- H˙ = − H2 − 2 2 a2 ciple to assume a new origin of gravity as an entropic √ 3/2 α µ 1 + 3ω k force. We have used this extension to calculate correc- + H2 + : (71) 3 2(2 + 3ω) a2 tions to Newton’s law of gravitation as well as the Fried- mann equations arising from the generalized uncertainty 3.3. Solutions of modified Raychaudhuri principle suggested by different approaches to quantum equation gravity such as string theory and black hole physics. We followed the following procedure: modified theory of gravity → modified black hole entropy→ modified holographic Here we discuss solutions of the modified Raychaudhuri surface entropy → Newton’s law corrections→ modified equation (72) for the flat case, k = 0. In this case, equa- Friedmann equations tion (71) has the form . √ We found that the corrections for Newton’s law of grav- 3(1 + ω) α µ 1 + 3ω ity agree with the brane world corrections. This suggests H˙ = − H2 + H3: (72) 2 3 2(2 + 3ω) that GUP and brane world may have very similar gravity

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at low-energy. Besides, we note that the derived cor- tions of these modified Friedmann’s equations on specific rection terms for the Friedmann equation vanishes rapidly inflationary models as well as the reheating phase of the with increasing apparent horizon radius, as expected. This universe. We hope to report on these issues soon. means that the corrections become relevant at the early universe, in particular, with the inflationary models where the physical scales are few orders of magnitude less than Acknowledgments the Planck scale. When the universe becomes large, these corrections can be ignored and the modified Friedmann The research of AFA is supported by Benha University equation reduces to the standard Friedman equation. We (www.bu.edu.eg) and CFP in Zewail City. AFA would like a t can understand that when ( ) is large, it is difficult to ex- to thank ICTP, Trieste for the kind hospitality where the cite these modes and hence, the low-energy modes domi- present work was finished. The authors thank the anony- nates the entropy. But at the early universe, a large num- mous referees for useful comments and suggestions. ber of excited modes can contribute causing a modification to the area law leading to the modified Friedmann equations. References But could we observe the impact of these corrections on the early universe? Since these corrections modify the [1] D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B standard FRW cosmology, especially in early times, it is 216, 41 (1989) expected that they have some consequences on inflation. [2] M. Maggiore, Phys. Lett. B 304, 65 (1993) One of the interesting results reported in the Planck 2013 [3] M. Maggiore, Phys. Rev. D 49, 5182 (1994) data [77] is that exact scale-invariance of the scalar power [4] M. Maggiore, Phys. Lett. B 319, 83 (1993) spectrum has been ruled out by more than 5σ. Meaning [5] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995) that, the early universe tiny quantum fluctuations, which [6] S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, eventually cause the formation of galaxies, not only de- S. Scherer, H. Stoecker, Phys. Lett. B 575, 85 (2003) pend on the mode wave number k, but also on some phys- [7] C. Bambi, F. R. Urban, Class. Quant. Grav. 25, 095006 ical scale. This shows that scalar power spectrum and (2008) other inflation parameters could depend on physical scale. [8] F. Scardigli, Phys. Lett. B 452, 39 (1999) The energy scale of inflation models has to be around [9] A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D52, Grand Unified Theories (GUT) scale or larger, therefore, 1108 (1995) this cutoff could be the Planck scale. [10] A. Kempf, J. Phys. A 30, 2093 (1997) There are two distinct possibilities to introduce the Planck [11] F. Brau, J. Phys. A 32, 7691 (1999) scale to modify the standard inflation scenario. The first [12] A. F. Ali, S. Das, E. C. Vagenas, Phys. Lett. B 678, possibility is to use it as a momentum-cutoff in the quan- 497 (2009) tum field theory of the inflaton field (see for example [79] [13] S. Das, E. C. Vagenas, A. F. Ali, Phys. Lett. B 690, and references there in). The second possibility is to mod- 407 (2010) ify general relativity, which will lead to a modified Fried- [14] G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 mann equation. The latter framework is considered in sev- (2002) eral interesting inflationary models, such as Brane-world, [15] G. Amelino-Camelia, Phys. Lett. B 510, 255 (2001) f R and ( ) inflationary models. Consequences of modifying [16] J. Kowalski-Glikman, S. Nowak, Phys. Lett. B 539, Friedmann’s equation on inflation have been discussed in 126 (2002) literature, for example see [78] and references there in. [17] J. Magueijo, L. Smolin, Phys. Rev. D 67, 044017 Our framework lies in the second class which modifies (2003) the Hubble parameter H as a function of time compared [18] J. Magueijo, L. Smolin, Class. Quant. Grav. 21, 1725 to that of the standard FRW cosmology, as expressed in (2004) equations (53) and (74). Since the slow-rolling parame- [19] J. L. Cortes, J. Gamboa, Phys. Rev. D71, 065015 ters and η depend on H, changes in H will affect them (2005) and the scalar spectral index ns, which will have an im- [20] S. Das, E. C. Vagenas, Phys. Rev. Lett. 101, 221301 pact on observations. The coming generations of CMB (2008) precise observations will be able to measure the inflation [21] S. Das, E. C. Vagenas, Can. J. Phys. 87, 233 (2009) parameters with high accuracy, therefore enabling differ- [22] A. F. Ali, S. Das, E. C. Vagenas, Phys. Rev. D 84, ent inflationary models to be distinguished. In the future, 044013 (2011) it would be interesting to apply our approach to investiga- [23] W. Chemissany, S. Das, A. F. Ali, E. C. Vagenas, JCAP

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